Conformal Regge theory (1209.4355v5)

Abstract: We generalize Regge theory to correlation functions in conformal field theories. This is done by exploring the analogy between Mellin amplitudes in AdS/CFT and S-matrix elements. In the process, we develop the conformal partial wave expansion in Mellin space, elucidating the analytic structure of the partial amplitudes. We apply the new formalism to the case of four point correlation functions between protected scalar operators in N=4 Super Yang Mills, in cases where the Regge limit is controlled by the leading twist operators associated to the pomeron-graviton Regge trajectory. At weak coupling, we are able to predict to arbitrary high order in the 't Hooft coupling the behaviour near J=1 of the OPE coefficients C_{OOJ} between the external scalars and the spin J leading twist operators. At strong coupling, we use recent results for the anomalous dimension of the leading twist operators to improve current knowledge of the AdS graviton Regge trajectory - in particular, determining the next and next to next leading order corrections to the intercept. Finally, by taking the flat space limit and considering the Virasoro-Shapiro S-matrix element, we compute the strong coupling limit of the OPE coefficient C_{LLJ} between two Lagrangians and the leading twist operators of spin J.

• The paper generalizes Regge theory to conformal field theories by mapping Mellin amplitudes to scattering processes, enhancing our understanding of high-energy limits.
• It develops a precise conformal partial wave expansion in Mellin space that clarifies the analytic structure of conformal blocks.
• The research applies its framework to SYM theory, predicting operator behaviors and refining insights into the AdS graviton Regge trajectory at strong coupling.

Insights into "Conformal Blocks in Mellin Space"

The research paper "Conformal Blocks in Mellin Space" by Miguel S. Costa, Vasco Goncalves, and Joao Penedones presents an advanced exploration of conformal field theory (CFT) within the Mellin representation framework. This work extends Regge theory by incorporating the concept into conformal field theories (CFTs) using Mellin amplitudes. The paper offers a meticulous analysis of the interplay between Mellin amplitudes in Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence and S-matrix elements, akin to scattering amplitudes in flat spacetime.

Core Contributions

The key contributions of the paper are as follows:

1. Generalization of Regge Theory: The authors adapt Regge theory to CFT by exploring similarities between Mellin amplitudes and traditional scattering amplitudes. This extension is crucial for understanding high-energy scattering processes involving an infinite number of resonances, which are characteristic of quantum chromodynamics (QCD) and string theory.
2. Conformal Partial Wave Expansion: A detailed development of the conformal partial wave expansion in Mellin space is presented. The work elucidates the analytic structure of partial amplitudes, which is instrumental in understanding the conformal blocks' behavior.
3. Application to SYM Theory: The formalism is applied to four-point correlation functions between scalar operators in ${N}=4$ Super Yang-Mills (SYM) theory. The analysis focuses on cases controlled by pomeron-graviton Regge trajectories, predicting behaviors of operators near $J=1$ at weak coupling.
4. Strong Coupling Analysis: At strong coupling, the research utilizes recent findings on anomalous dimensions to refine knowledge of the AdS graviton Regge trajectory. The paper identifies corrections to the intercept, offering improved insights into the interactions of massive string states in AdS.
5. Implications for Flat Space Limit: By taking the flat space limit and considering the Virasoro-Shapiro S-matrix element, the paper calculates the strong coupling limit of the OPE coefficient $C_{ LL}J}$, which pertains to interactions between Lagrangian operators and leading twist operators of spin $J$.

Theoretical and Practical Implications

1. Theoretical Framework: The generalization of Regge theory to CFT settings provides a robust framework for examining high-energy limits in theoretical physics. This contributes to a deeper understanding of AdS/CFT correspondence, crucial for theoretical explorations in string theory and beyond.
2. Predictive Power: The research leverages the Mellin space representation to predict operator behavior in SYM theory. These predictions, centered around perturbative and non-perturbative regimes, enhance our understanding of gauge theories at varying coupling strengths.
3. Future of AdS/CFT: The methodology adopted in this paper lays the groundwork for subsequent explorations into multi-point correlation functions within large N gauge theories. These insights can potentially pave the way for novel computational techniques in higher-dimensional CFTs and related fields.

Future Directions

The paper's outcomes suggest several avenues for future research:

• Multi-Trajectory Studies: Examination of other Regge trajectories with distinct quantum numbers could yield new insights into the operator spectrum of SYM theories.
• Higher-Order Perturbative Analyses: The conformal Regge theory framework may be explored at higher perturbation levels, capitalizing on available data for further insights into the dynamics of gauge theories.
• Interplay with Flat Space Scattering: Investigating the transition from conformal field theory to confining gauge theories could deepen the understanding of how continuous Mellin variables translate to discrete ones in standard Regge frameworks.

Overall, "Conformal Blocks in Mellin Space" provides a significant leap in understanding conformal field theories via Mellin representations, bridging critical gaps in the treatment of high-energy physics processes.